3.121 \(\int \frac{(a+b x)^{10} (A+B x)}{x^5} \, dx\)

Optimal. Leaf size=215 \[ -\frac{a^{10} A}{4 x^4}-\frac{a^9 (a B+10 A b)}{3 x^3}-\frac{5 a^8 b (2 a B+9 A b)}{2 x^2}-\frac{15 a^7 b^2 (3 a B+8 A b)}{x}+30 a^6 b^3 \log (x) (4 a B+7 A b)+42 a^5 b^4 x (5 a B+6 A b)+21 a^4 b^5 x^2 (6 a B+5 A b)+10 a^3 b^6 x^3 (7 a B+4 A b)+\frac{15}{4} a^2 b^7 x^4 (8 a B+3 A b)+\frac{1}{6} b^9 x^6 (10 a B+A b)+a b^8 x^5 (9 a B+2 A b)+\frac{1}{7} b^{10} B x^7 \]

[Out]

-(a^10*A)/(4*x^4) - (a^9*(10*A*b + a*B))/(3*x^3) - (5*a^8*b*(9*A*b + 2*a*B))/(2*
x^2) - (15*a^7*b^2*(8*A*b + 3*a*B))/x + 42*a^5*b^4*(6*A*b + 5*a*B)*x + 21*a^4*b^
5*(5*A*b + 6*a*B)*x^2 + 10*a^3*b^6*(4*A*b + 7*a*B)*x^3 + (15*a^2*b^7*(3*A*b + 8*
a*B)*x^4)/4 + a*b^8*(2*A*b + 9*a*B)*x^5 + (b^9*(A*b + 10*a*B)*x^6)/6 + (b^10*B*x
^7)/7 + 30*a^6*b^3*(7*A*b + 4*a*B)*Log[x]

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Rubi [A]  time = 0.4691, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{a^{10} A}{4 x^4}-\frac{a^9 (a B+10 A b)}{3 x^3}-\frac{5 a^8 b (2 a B+9 A b)}{2 x^2}-\frac{15 a^7 b^2 (3 a B+8 A b)}{x}+30 a^6 b^3 \log (x) (4 a B+7 A b)+42 a^5 b^4 x (5 a B+6 A b)+21 a^4 b^5 x^2 (6 a B+5 A b)+10 a^3 b^6 x^3 (7 a B+4 A b)+\frac{15}{4} a^2 b^7 x^4 (8 a B+3 A b)+\frac{1}{6} b^9 x^6 (10 a B+A b)+a b^8 x^5 (9 a B+2 A b)+\frac{1}{7} b^{10} B x^7 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^10*(A + B*x))/x^5,x]

[Out]

-(a^10*A)/(4*x^4) - (a^9*(10*A*b + a*B))/(3*x^3) - (5*a^8*b*(9*A*b + 2*a*B))/(2*
x^2) - (15*a^7*b^2*(8*A*b + 3*a*B))/x + 42*a^5*b^4*(6*A*b + 5*a*B)*x + 21*a^4*b^
5*(5*A*b + 6*a*B)*x^2 + 10*a^3*b^6*(4*A*b + 7*a*B)*x^3 + (15*a^2*b^7*(3*A*b + 8*
a*B)*x^4)/4 + a*b^8*(2*A*b + 9*a*B)*x^5 + (b^9*(A*b + 10*a*B)*x^6)/6 + (b^10*B*x
^7)/7 + 30*a^6*b^3*(7*A*b + 4*a*B)*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{10}}{4 x^{4}} + \frac{B b^{10} x^{7}}{7} - \frac{a^{9} \left (10 A b + B a\right )}{3 x^{3}} - \frac{5 a^{8} b \left (9 A b + 2 B a\right )}{2 x^{2}} - \frac{15 a^{7} b^{2} \left (8 A b + 3 B a\right )}{x} + 30 a^{6} b^{3} \left (7 A b + 4 B a\right ) \log{\left (x \right )} + 210 a^{5} b^{4} x \left (\frac{6 A b}{5} + B a\right ) + 42 a^{4} b^{5} \left (5 A b + 6 B a\right ) \int x\, dx + 10 a^{3} b^{6} x^{3} \left (4 A b + 7 B a\right ) + \frac{15 a^{2} b^{7} x^{4} \left (3 A b + 8 B a\right )}{4} + a b^{8} x^{5} \left (2 A b + 9 B a\right ) + \frac{b^{9} x^{6} \left (A b + 10 B a\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**10*(B*x+A)/x**5,x)

[Out]

-A*a**10/(4*x**4) + B*b**10*x**7/7 - a**9*(10*A*b + B*a)/(3*x**3) - 5*a**8*b*(9*
A*b + 2*B*a)/(2*x**2) - 15*a**7*b**2*(8*A*b + 3*B*a)/x + 30*a**6*b**3*(7*A*b + 4
*B*a)*log(x) + 210*a**5*b**4*x*(6*A*b/5 + B*a) + 42*a**4*b**5*(5*A*b + 6*B*a)*In
tegral(x, x) + 10*a**3*b**6*x**3*(4*A*b + 7*B*a) + 15*a**2*b**7*x**4*(3*A*b + 8*
B*a)/4 + a*b**8*x**5*(2*A*b + 9*B*a) + b**9*x**6*(A*b + 10*B*a)/6

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Mathematica [A]  time = 0.187994, size = 210, normalized size = 0.98 \[ -\frac{a^{10} (3 A+4 B x)}{12 x^4}-\frac{5 a^9 b (2 A+3 B x)}{3 x^3}-\frac{45 a^8 b^2 (A+2 B x)}{2 x^2}-\frac{120 a^7 A b^3}{x}+30 a^6 b^3 \log (x) (4 a B+7 A b)+210 a^6 b^4 B x+126 a^5 b^5 x (2 A+B x)+35 a^4 b^6 x^2 (3 A+2 B x)+10 a^3 b^7 x^3 (4 A+3 B x)+\frac{9}{4} a^2 b^8 x^4 (5 A+4 B x)+\frac{1}{3} a b^9 x^5 (6 A+5 B x)+\frac{1}{42} b^{10} x^6 (7 A+6 B x) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^10*(A + B*x))/x^5,x]

[Out]

(-120*a^7*A*b^3)/x + 210*a^6*b^4*B*x + 126*a^5*b^5*x*(2*A + B*x) - (45*a^8*b^2*(
A + 2*B*x))/(2*x^2) + 35*a^4*b^6*x^2*(3*A + 2*B*x) - (5*a^9*b*(2*A + 3*B*x))/(3*
x^3) + 10*a^3*b^7*x^3*(4*A + 3*B*x) - (a^10*(3*A + 4*B*x))/(12*x^4) + (9*a^2*b^8
*x^4*(5*A + 4*B*x))/4 + (a*b^9*x^5*(6*A + 5*B*x))/3 + (b^10*x^6*(7*A + 6*B*x))/4
2 + 30*a^6*b^3*(7*A*b + 4*a*B)*Log[x]

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Maple [A]  time = 0.011, size = 240, normalized size = 1.1 \[{\frac{{b}^{10}B{x}^{7}}{7}}+{\frac{A{x}^{6}{b}^{10}}{6}}+{\frac{5\,B{x}^{6}a{b}^{9}}{3}}+2\,A{x}^{5}a{b}^{9}+9\,B{x}^{5}{a}^{2}{b}^{8}+{\frac{45\,A{x}^{4}{a}^{2}{b}^{8}}{4}}+30\,B{x}^{4}{a}^{3}{b}^{7}+40\,A{x}^{3}{a}^{3}{b}^{7}+70\,B{x}^{3}{a}^{4}{b}^{6}+105\,A{x}^{2}{a}^{4}{b}^{6}+126\,B{x}^{2}{a}^{5}{b}^{5}+252\,Ax{a}^{5}{b}^{5}+210\,Bx{a}^{6}{b}^{4}+210\,A\ln \left ( x \right ){a}^{6}{b}^{4}+120\,B\ln \left ( x \right ){a}^{7}{b}^{3}-{\frac{45\,{a}^{8}{b}^{2}A}{2\,{x}^{2}}}-5\,{\frac{{a}^{9}bB}{{x}^{2}}}-120\,{\frac{{a}^{7}{b}^{3}A}{x}}-45\,{\frac{{a}^{8}{b}^{2}B}{x}}-{\frac{10\,{a}^{9}bA}{3\,{x}^{3}}}-{\frac{{a}^{10}B}{3\,{x}^{3}}}-{\frac{A{a}^{10}}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^10*(B*x+A)/x^5,x)

[Out]

1/7*b^10*B*x^7+1/6*A*x^6*b^10+5/3*B*x^6*a*b^9+2*A*x^5*a*b^9+9*B*x^5*a^2*b^8+45/4
*A*x^4*a^2*b^8+30*B*x^4*a^3*b^7+40*A*x^3*a^3*b^7+70*B*x^3*a^4*b^6+105*A*x^2*a^4*
b^6+126*B*x^2*a^5*b^5+252*A*x*a^5*b^5+210*B*x*a^6*b^4+210*A*ln(x)*a^6*b^4+120*B*
ln(x)*a^7*b^3-45/2*a^8*b^2/x^2*A-5*a^9*b/x^2*B-120*a^7*b^3/x*A-45*a^8*b^2/x*B-10
/3*a^9/x^3*A*b-1/3*a^10/x^3*B-1/4*a^10*A/x^4

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Maxima [A]  time = 1.41114, size = 324, normalized size = 1.51 \[ \frac{1}{7} \, B b^{10} x^{7} + \frac{1}{6} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{6} +{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{5} + \frac{15}{4} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{4} + 10 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{3} + 21 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{2} + 42 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x + 30 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} \log \left (x\right ) - \frac{3 \, A a^{10} + 180 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 30 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 4 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^5,x, algorithm="maxima")

[Out]

1/7*B*b^10*x^7 + 1/6*(10*B*a*b^9 + A*b^10)*x^6 + (9*B*a^2*b^8 + 2*A*a*b^9)*x^5 +
15/4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^4 + 10*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^3 + 21*
(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^2 + 42*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x + 30*(4*B*a^7
*b^3 + 7*A*a^6*b^4)*log(x) - 1/12*(3*A*a^10 + 180*(3*B*a^8*b^2 + 8*A*a^7*b^3)*x^
3 + 30*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 4*(B*a^10 + 10*A*a^9*b)*x)/x^4

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Fricas [A]  time = 0.209535, size = 331, normalized size = 1.54 \[ \frac{12 \, B b^{10} x^{11} - 21 \, A a^{10} + 14 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 84 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 315 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 840 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 1764 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 3528 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 2520 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} \log \left (x\right ) - 1260 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 210 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 28 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{84 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^5,x, algorithm="fricas")

[Out]

1/84*(12*B*b^10*x^11 - 21*A*a^10 + 14*(10*B*a*b^9 + A*b^10)*x^10 + 84*(9*B*a^2*b
^8 + 2*A*a*b^9)*x^9 + 315*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^8 + 840*(7*B*a^4*b^6 + 4
*A*a^3*b^7)*x^7 + 1764*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^6 + 3528*(5*B*a^6*b^4 + 6*A
*a^5*b^5)*x^5 + 2520*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^4*log(x) - 1260*(3*B*a^8*b^2
+ 8*A*a^7*b^3)*x^3 - 210*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 - 28*(B*a^10 + 10*A*a^9*b
)*x)/x^4

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Sympy [A]  time = 6.98476, size = 243, normalized size = 1.13 \[ \frac{B b^{10} x^{7}}{7} + 30 a^{6} b^{3} \left (7 A b + 4 B a\right ) \log{\left (x \right )} + x^{6} \left (\frac{A b^{10}}{6} + \frac{5 B a b^{9}}{3}\right ) + x^{5} \left (2 A a b^{9} + 9 B a^{2} b^{8}\right ) + x^{4} \left (\frac{45 A a^{2} b^{8}}{4} + 30 B a^{3} b^{7}\right ) + x^{3} \left (40 A a^{3} b^{7} + 70 B a^{4} b^{6}\right ) + x^{2} \left (105 A a^{4} b^{6} + 126 B a^{5} b^{5}\right ) + x \left (252 A a^{5} b^{5} + 210 B a^{6} b^{4}\right ) - \frac{3 A a^{10} + x^{3} \left (1440 A a^{7} b^{3} + 540 B a^{8} b^{2}\right ) + x^{2} \left (270 A a^{8} b^{2} + 60 B a^{9} b\right ) + x \left (40 A a^{9} b + 4 B a^{10}\right )}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**10*(B*x+A)/x**5,x)

[Out]

B*b**10*x**7/7 + 30*a**6*b**3*(7*A*b + 4*B*a)*log(x) + x**6*(A*b**10/6 + 5*B*a*b
**9/3) + x**5*(2*A*a*b**9 + 9*B*a**2*b**8) + x**4*(45*A*a**2*b**8/4 + 30*B*a**3*
b**7) + x**3*(40*A*a**3*b**7 + 70*B*a**4*b**6) + x**2*(105*A*a**4*b**6 + 126*B*a
**5*b**5) + x*(252*A*a**5*b**5 + 210*B*a**6*b**4) - (3*A*a**10 + x**3*(1440*A*a*
*7*b**3 + 540*B*a**8*b**2) + x**2*(270*A*a**8*b**2 + 60*B*a**9*b) + x*(40*A*a**9
*b + 4*B*a**10))/(12*x**4)

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GIAC/XCAS [A]  time = 0.255071, size = 325, normalized size = 1.51 \[ \frac{1}{7} \, B b^{10} x^{7} + \frac{5}{3} \, B a b^{9} x^{6} + \frac{1}{6} \, A b^{10} x^{6} + 9 \, B a^{2} b^{8} x^{5} + 2 \, A a b^{9} x^{5} + 30 \, B a^{3} b^{7} x^{4} + \frac{45}{4} \, A a^{2} b^{8} x^{4} + 70 \, B a^{4} b^{6} x^{3} + 40 \, A a^{3} b^{7} x^{3} + 126 \, B a^{5} b^{5} x^{2} + 105 \, A a^{4} b^{6} x^{2} + 210 \, B a^{6} b^{4} x + 252 \, A a^{5} b^{5} x + 30 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{3 \, A a^{10} + 180 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 30 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 4 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10/x^5,x, algorithm="giac")

[Out]

1/7*B*b^10*x^7 + 5/3*B*a*b^9*x^6 + 1/6*A*b^10*x^6 + 9*B*a^2*b^8*x^5 + 2*A*a*b^9*
x^5 + 30*B*a^3*b^7*x^4 + 45/4*A*a^2*b^8*x^4 + 70*B*a^4*b^6*x^3 + 40*A*a^3*b^7*x^
3 + 126*B*a^5*b^5*x^2 + 105*A*a^4*b^6*x^2 + 210*B*a^6*b^4*x + 252*A*a^5*b^5*x +
30*(4*B*a^7*b^3 + 7*A*a^6*b^4)*ln(abs(x)) - 1/12*(3*A*a^10 + 180*(3*B*a^8*b^2 +
8*A*a^7*b^3)*x^3 + 30*(2*B*a^9*b + 9*A*a^8*b^2)*x^2 + 4*(B*a^10 + 10*A*a^9*b)*x)
/x^4